A computer will be required for the remaining problems in this section. In Problems 9 through 12, an initial value problem and its exact solution are given. In each of these four problems, use the Runge–Kutta method with step sizes h = 0.1 and h = 0.05 to approximate to five decimal places the values x(1) and y(1). Compare the approximations with the actual values.

x' = 2xy, x(0) = 1, y(0) = 0; y' = x + 2y, x (t) = et cost, y(t) = et sint